Problem 4 - Classical Mechanics - An Undergraduate Text

Question

A particle \(P\) of mass \(m\) moves under the simple harmonic field \(\boldsymbol{F}=-\left(m \Omega^{2} r\right) \widehat{\boldsymbol{r}}\), where \(\Omega\) is a positive constant. Obtain the radial motion equation and show that all orbits of \(P\) are bounded. Initially \(P\) is at a point \(C\), a distance \(c\) from \(O\), when it is projected with speed \(\Omega c\) in a direction making an acute angle \(\alpha\) with \(O C\). Find the equation satisfied by the apsidal distances. Given that the orbit of \(P\) is an ellipse with centre \(O\), find the semi-major and semiminor axes of this ellipse.

Step-by-Step Solution

Answer

The radial motion equation is \(r'' + Ω^2r = 0\), proving that the orbits are bounded. The equation satisfied by apsidal distances in the orbit is \(r = c cosα/(1- sinα)\). The semi-major axis of the ellipse is \(3c/4\) and the semi-minor axis is \(c/4\).

Step 1: Finding the radial motion equation

Starting with the given force field equation \(F = -mΩ^2r\), we can derive the radial motion equation using Newton's second law of motion, which is \(F = ma\). But the acceleration (a) here can be represented as \(r''\) (double derivative of r with respect to time). So, we get the equation \(F = m r''\). Replacing \(F = -mΩ^2r\) into this, we get the radial equation of motion: \(r'' + Ω^2r = 0\).

Step 2: Show that orbits are bounded

The given radial motion equation is the equation of simple harmonic motion, which implies that the motion is oscillatory and bounded. In simple harmonic motion, the particles always oscillate between two extreme positions; hence, the motion is always bounded.

Step 3: Find the apsidal distances equation

Given that the particle is projected with speed \(Ωc\) making an angle \(α\) with OC, we can find equation for apsidal distances using Kinetic energy = Potential energy. So, \(\frac{1}{2} m v^2 = \frac{1}{2} m Ω^2 r^2\). Simplifying this, we find \(v = Ω r\). Given that v = \(Ω c cosα + Ω r sinα\), replacing v from the kinetic energy equation gives us the apsidal distance equation: \(r = c cosα/(1- sinα)\)

Step 4: Obtain semi-major and semi-minor axes of the ellipse

Given that the orbit of P is an ellipse centered at O, and from the apsidal distances equation where \(r = r_{min}\) for \(α = 0\) and \(r = r_{max}\) for \(α = π/2\), we can find the semi-major and semi-minor axes which are \((r_{max} + r_{min})/2\) and \((r_{max} - r_{min})/2\) respectively. For \(\alpha = 0\), \(r = c\) and for \(\alpha = pi/2\), \(r = c/2\). Thus semi-major and semi-minor axes are \(3c/4\) and \(c/4\) respectively.

Key Concepts

Radial Motion Equation

The radial motion equation is a fundamental part of understanding how a particle moves in a radial force field. Here, we start with the force acting on the particle given by the equation \( F = -m \Omega^2 r \), where \( \Omega \) is a constant. To find the radial motion equation, we apply Newton's second law, \( F = ma \), where acceleration \( a \) is \( r'' \) (the second derivative of \( r \) with respect to time). By replacing \( F \) in the equation, we get \( m r'' = -m \Omega^2 r \).
This leads to a simplified radial motion equation \( r'' + \Omega^2 r = 0 \). This equation is recognizable as the equation of simple harmonic motion. Its solutions describe oscillations, meaning that the particle moves back and forth in a regular manner.

Apsidal Distances

Apsidal distances refer to the farthest and closest points that a particle can reach when moving along a path, such as an ellipse. They are found using energy conservation methods. When a particle moves under force, its kinetic and potential energies exchange, but the total remains constant. In this scenario, the kinetic and potential energies are equated:
\[ \frac{1}{2} m v^2 = \frac{1}{2} m \Omega^2 r^2 \]
Simplifying, we get \( v = \Omega r \). Given a projected speed of \( \Omega c \) at angle \( \alpha \), the velocity equation becomes \( v = \Omega c \cos{\alpha} + \Omega r \sin{\alpha} \). Replacing \( v \) gives us the apsidal distance equation:
\[ r = \frac{c \cos{\alpha}}{1 - \sin{\alpha}} \]
This equation helps us identify minimum and maximum distances \( r_{min} \) and \( r_{max} \) during motion.

Bounded Orbits

Bounded orbits refer to paths on which the particle consistently oscillates back and forth rather than drifting off indefinitely. The radial motion equation \( r'' + \Omega^2 r = 0 \) indicates that the motion is simple harmonic. Simple harmonic motion suggests that the particle's movement is confined between two limits, indicating that the orbit remains bounded.
A basic characteristic of bounded motion is its periodicity - the particle revisits its positions in a cyclic manner. These properties ensure that all orbits described by this equation will loop continuously without escaping to infinity.

Ellipse Properties

In the context of this problem, understanding ellipse properties helps illustrate the particle's motion. An ellipse is a closed curve defined by its semi-major and semi-minor axes. For elliptical orbits, these axes can be derived from apsidal distances. Using the found points \( r_{max} \) and \( r_{min} \), the semi-major axis (the longest radius) is the average \((r_{max} + r_{min})/2\), and the semi-minor axis (shortest radius) is \((r_{max} - r_{min})/2\).
In this scenario, the semi-major axis is \( 3c/4 \) and the semi-minor axis is \( c/4 \). This aligns with the general characteristics of an ellipse, where the center point is equidistant from the opposite, extreme orbit points. Understanding these allow you to visualize particle movements around the center, ensuring comprehension of the geometric path.